General solution of $u_t + u_x = -u^2$

Question

Can someone help me to write down a formula for the general solution to the nonlinear partial differential equation $$u_t + u_x + u^2 = 0$$ and how do I show that if the initial data is positive and bounded, $0 ≤ u(0,x) = f(x) ≤ M$, then the solution exists for all $t > 0$, and $u(t,x) → 0$ as $t → ∞$

Answer 1

You get to solve $$ \frac{dt}1=\frac{dx}1=-\frac{du}{u^2} $$ leading to $$x-t=c_1, ~~u^{-1}-t=c_2$$ along the characteristic curves. As the characteristic curves inside the solution surface form a one-parameter family, there is a function $c_2=\phi(c_1)$ connecting the integration constants, thus $$ u(t,x)=\frac1{t+\phi(x-t)}. $$ Now apply all the other assumptions and try to prove the claims.

As $f(x)=u(0,x)=\frac{1}{\phi(t)}$, the solution can be rewritten as $$ u(t,x)=\frac{f(x-t)}{1+t\,f(x-t)}. $$

If $f(x_*)<0$ at some point, then select $t_*=-\frac1{f(x_*)}$ and consider the point $(t,x)=(t_*, x_*+t_*)$. The denominator will be zero, there will be a singularity at that point.